How many matrices `X` with entries `{0,1,2}` are there for which sum of diagonal entries of `X.X^(T)` is 7?

Options

A. 560
B. 562
C. 504
D. 540

Answer

D

Explanation

`:'a_(1)^(2)+b_(1)^(2)+c_(1)^(2)+a_(2)^(2)+b_(2)^(2)+c_(2)^(2)+a_(3)^(2)+b_(3)^(2)+c_(3)^(2)=7`, where `langlea_(i),b_(i),c_(i)rangle` are elements of `i^(th)` row.
`:.` Case I: All seven entries are 1 and rest of all zero
`.^(9)C_(7)=.^(9)C_(2)=36`
Case II: Only one entry is 2 & three entries are 1 and rest of all 0's
`.^(9)C_(4)xx.^(4)C_(1)=504`
`:.` Total `=540`